# Eigenvalues of matrix product with a diagonal matrix

I have the following problem: Suppose we have a nonnegative diagonal matrix $$A \in \mathbb{R}_+^{n\times n}$$ and a matrix $$B \in \mathbb{R}^{n\times n}$$ with $$Re(\lambda_i(B)) \leq 0,\; i=1,\dots,n$$

where $$\lambda_i(B)$$ denotes the $$i$$-th eigenvalue of $$B$$ and $$Re(\cdot)$$ denotes the real part.

Is it possible to show that $$Re(\lambda_i(AB))\leq 0$$ for $$i=1,\dots,n$$?

I found similar questions but with different conditions on $$A$$ and $$B$$:

Any help is appreciated!

$$A=\begin{bmatrix}4 & 0\\0 & 1\end{bmatrix}, B=\begin{bmatrix}1 & -2\\2 & -2\end{bmatrix}$$
For $$B$$ we have $$Re(\lambda_{1}(B)) = Re(\lambda_{2}(B)) =-\frac{1}{2}$$ and for $$AB = \begin{bmatrix}4 & -8\\2 & -2\end{bmatrix}$$ it is $$Re(\lambda_{1}(AB)) = Re(\lambda_{2}(AB)) = 1$$.
• Do you mind $\lambda_{1,2}$? – user376343 Apr 4 at 8:23